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Properties

Label 1-896-896.205-r0-0-0
Degree $1$
Conductor $896$
Sign $0.597 + 0.801i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0654 + 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (−0.442 − 0.896i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (0.946 − 0.321i)19-s + (0.130 + 0.991i)23-s + (−0.130 + 0.991i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (0.751 − 0.659i)37-s + ⋯
L(s)  = 1  + (0.0654 + 0.997i)3-s + (−0.659 − 0.751i)5-s + (−0.991 + 0.130i)9-s + (−0.442 − 0.896i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (0.946 − 0.321i)19-s + (0.130 + 0.991i)23-s + (−0.130 + 0.991i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (0.751 − 0.659i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.597 + 0.801i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.597 + 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9035531883 + 0.4536363140i\)
\(L(\frac12)\) \(\approx\) \(0.9035531883 + 0.4536363140i\)
\(L(1)\) \(\approx\) \(0.8537187614 + 0.1836358066i\)
\(L(1)\) \(\approx\) \(0.8537187614 + 0.1836358066i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0654 + 0.997i)T \)
5 \( 1 + (-0.659 - 0.751i)T \)
11 \( 1 + (-0.442 - 0.896i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (0.946 - 0.321i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (0.555 + 0.831i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (0.751 - 0.659i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.831 + 0.555i)T \)
47 \( 1 + (-0.258 + 0.965i)T \)
53 \( 1 + (0.442 + 0.896i)T \)
59 \( 1 + (0.321 - 0.946i)T \)
61 \( 1 + (0.896 + 0.442i)T \)
67 \( 1 + (-0.0654 - 0.997i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.608 - 0.793i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (0.793 + 0.608i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.25404630281812716088576676961, −20.81529086070574642084683047696, −20.06692593921588743723446226902, −19.40193944165668162893060557461, −18.68490939774732850066744299325, −17.95906919815071285246182816903, −17.37742461055526662856051860423, −16.23096963976368381501205188261, −15.25974964931344056575194474107, −14.63271712248378330617743300838, −13.81191423890137495593036591462, −12.92668468167796179871125041544, −12.01165015448557529900304366074, −11.60917933742673784973602968130, −10.489358866277778481329348339032, −9.6201573243794519510349121079, −8.40033769743451848778690455785, −7.616873654356272940965213077845, −7.03181206747961486798538997629, −6.30529174951839013231551040383, −5.0271720055656614603840435464, −4.05125954074230678658246168326, −2.6040622397548905053032809939, −2.33989159194066580961147559425, −0.61238907645712057211192636846, 0.84642582737449755003499587484, 2.623612015376443221075747401455, 3.433362580202668725406508524712, 4.49365052740043416973127057010, 5.06996903513630753774799359112, 5.96927151738344263846815320588, 7.4157691812246151099356480104, 8.19517535344329666198152085962, 9.04835324368191304693495949679, 9.66240428517926990992270672322, 10.82213687266149322878832111426, 11.371457007638338530743892925570, 12.2890512945422678904277694802, 13.25052301586128098128079265557, 14.14538518417943272594979097262, 15.062634301917334168392241630407, 15.98392422944708040795919618151, 16.094367470068379951991990939513, 17.23359634887637457290643220256, 17.85691795312082245461549893297, 19.36318206259942185211276542536, 19.691003703058575607903400768644, 20.47820120754053638091411479028, 21.37021256927141477279774154655, 21.88979080191012903224900071730

Graph of the $Z$-function along the critical line